Theorem pwpwuni 46213

Description: Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
pwpwuni	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))

Proof of Theorem pwpwuni

Step	Hyp	Ref	Expression
1		elpwg 4562	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵))
2		sspwuni 5062	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
4		uniexg 7764	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
5		elpwg 4562	. . . 4 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
6	4, 5	syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
7	6	bicomd 226	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))
8	1, 3, 7	3bitrd 308	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2146  Vcvv 3453   ⊆ wss 3899  𝒫 cpw 4559  ∪ cuni 4869

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3455  df-ss 3900  df-pw 4561  df-uni 4870

This theorem is used by:  psmeasurelem  47619